SOME RESULTS ON THE UNIQUE RANGE SETS

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J. Korean Math. Soc. 58 (2021), No. 3, pp. 741–760 https://doi.org/10.4134/JKMS.j200235

pISSN: 0304-9914 / eISSN: 2234-3008

Bikash Chakraborty, Jayanta Kamila, Amit Kumar Pal, and Sudip Saha

Abstract. In this paper, we exhibit the equivalence between different notions of unique range sets, namely, unique range sets, weighted unique range sets and weak-weighted unique range sets under certain conditions. Also, we present some uniqueness theorems which show how two mero-morphic functions are uniquely determined by their two finite shared sets. Moreover, in the last section, we make some observations that help us to construct other new classes of unique range sets.

1. Introduction: Unique range sets

We use M (C) to denote the field of all meromorphic functions. Let f ∈ M (C) and S ⊂ C ∪ {∞} be a non-empty set with distinct elements. We set

Ef(S) = [

a∈S

{z : f (z) − a = 0},

where a zero of f − a with multiplicity m counts m times in Ef(S). Let Ef(S) denote the collection of distinct elements in Ef(S).

Let g ∈ M (C). We say that two functions f and g share the set S CM (resp. IM) if Ef(S) = Eg(S) (resp. Ef(S) = Eg(S)).

In 1968, F. Gross ([14]) first studied the uniqueness problem of meromorphic functions that share distinct sets instead of values. Since then, the uniqueness theory of meromorphic functions under the set sharing environment has become one of the important branches in the value distribution theory.

In the same paper ([14]), F. Gross proved that there exist three finite sets Sj (j = 1, 2, 3) such that if two non-constant entire functions f and g share them, then f ≡ g. Later, in 1976, he asked the following question:

Received May 2, 2020; Revised September 24, 2020; Accepted February 16, 2021. 2010 Mathematics Subject Classification. Primary 30D30, 30D20, 30D35.

Key words and phrases. Unique range set, weighted set sharing, value distribution theory. The research work of the first and the second authors are supported by the Depart-ment of Higher Education, Science and Technology & Biotechnology, Govt. of West Bengal under the sanction order no. 216(sanc) /ST/P/S&T/16G-14/2018 dated 19/02/2019. The third and fourth authors are thankful to the Council of Scientific and Industrial Research, HRDG, India for granting Junior Research Fellowship (File No.: 09/106(0179)/2018-EMR-I & 08/525(0003)/2019-EMR-I respectively) during the tenure of which this work was done.

c

2021 Korean Mathematical Society

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Question 1.1 ([15]). Can one find two (or possible even one) finite sets Sj(j = 1, 2) such that if two non-constant entire functions f and g share them, then f ≡ g?

In 1982, F. Gross and C. C. Yang ([16]) first ensured the existence of such set. They proved that if two non-constant entire functions f and g share the set S = {z ∈ C : ez_{+ z = 0}, then f ≡ g.}

Moreover, this type of set was termed as a unique range set for entire func-tions. Later, similar definition for meromorphic functions was also introduced in the literature.

Definition 1.1 ([20]). Let S ⊂ C ∪ {∞}; f and g be two non-constant mero-morphic (resp. entire) functions. If Ef(S) = Eg(S) implies f ≡ g, then S is called a unique range set for meromorphic (resp. entire) functions or in brief URSM (resp. URSE).

Here, we note that the set provided by Gross and Yang ([16]) was an infinite set. Thus after the introduction to the idea of unique range sets, many efforts were made to seek unique range sets with cardinalities as small as possible.

In 1994, H. X. Yi ([22]), settled the question of Gross by exhibiting a unique range set for entire functions with 15 elements.

In the next year, P. Li and C. C. Yang ([19]) exhibited a unique range set for meromorphic (resp. entire) functions with 15 (resp. 7) elements. They considered the zero set of the following polynomial:

(1.1) P (z) = zn+ azn−m+ b,

where a and b are two non-zero constants such that zn+ azn−m+ b = 0 has no multiple roots. Also, m ≥ 2 (resp. 1), n > 2m + 10 (resp. 2m + 4) are integers with n and n − m having no common factors.

In 1996, H. X. Yi ([24]) further improved the result of Li and Yang ([19]) and obtained a unique range set for meromorphic functions with 13 elements. Also, in 2002, T. T. H. An ([2]) exhibited another new class of unique range set for meromorphic functions with 13 elements by considering the zero set of the following polynomial:

(1.2) P (z) = zn+ azn−m+ bzn−2m+ c,

where it was assumed that P (z) = 0 has no multiple roots and a, b, c ∈ C \ {0} such that a2_{6= 4b. Also, n and 2m are two positive integers such that n and} 2m have no common factors and n > 8 + 4m.

As an attempt to reduce the cardinality of the unique range sets, in 1998, G. Frank and M. Reinders ([11]) studied the zero set of the following polynomial and obtained a unique range set with 11 elements.

(1.3) P (z) =(n − 1)(n − 2) 2 z n_{− n(n − 2)z}n−1₊n(n − 1) 2 z n−2_{− c,} where n ≥ 11 and c 6= 0, 1.

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And, till today, this is the smallest available unique range set for meromor-phic functions. But, in 2007, T. C. Alzahary ([1]) exhibited another new class of unique range set for meromorphic functions with 11 elements by considering the zero set of the following polynomial:

(1.4) P (z) = azn− n(n − 1)z2_{+ 2n(n − 2)bz − (n − 1)(n − 2)b}2_,

where a and b are two non-zero complex numbers satisfying abn−26= 1, 2 and n ≥ 11.

Recently, another new class of unique range set for meromorphic functions with 11 elements were exhibited in ([7]) using the zero set of the following polynomial: (1.5) P (z) = zn− 2n n − mz n−m₊ n n − 2mz n−2m_{+ c,}

where c is any complex number satisfying |c| 6= _{(n−m)(n−2m)}2m2 and c 6= 0, c 6= −1−n−m2n +

n n−2m

2 and m ≥ 1, n > max{2m + 8, 4m + 1}.

Until now, the best results were given by H. Fujimoto ([12]) in 2000. He gave a generic unique range set for meromorphic functions of at least 11 elements when multiplicities are counted. To state those results, we need to explain some definitions.

Definition 1.2 ([12, 13]). Let P (z) be a non-constant monic polynomial. We call P (z) as a “uniqueness polynomial in broad sense” if P (f ) ≡ P (g) implies f ≡ g for any two non-constant meromorphic functions f, g; while a “unique-ness polynomial” if P (f ) ≡ cP (g) implies f ≡ g for any two non-constant meromorphic functions f, g and non-zero constant c.

For a discrete subset S = {a1, a2, . . . , an} ⊂ C (ai 6= aj), we consider the following polynomial

(1.6) P (z) = (z − a1)(z − a2) · · · (z − an).

Assume that the derivative P0(z) has k distinct zeros d1, d2, . . . , dk with mul-tiplicities q1, q2, . . . , qk respectively. Under the assumption that

(1.7) P (dls) 6= P (dlt) (1 ≤ ls< lt≤ k),

H. Fujimoto ([12]) proved the following theorem:

Theorem 1.1 ([12]). Let P (z) be a “uniqueness polynomial” of the form (1.6) satisfying the condition (1.7). Moreover, either k ≥ 3 or k = 2 and min{q1, q2} ≥ 2.

If S is the set of zeros of P (z), then S is a unique range set for meromorphic (resp. entire) function whenever n > 2k + 6 (resp. n > 2k + 2).

Remark 1.1. We note that Theorem 1.1 gives the best possible generic unique range set for meromorphic function when k = 2 (i.e., unique range set with 11 elements).

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In 2017, in ([8]), the form of the polynomial when k = 2 was illustrated in more general settings. If k = 2, then the unique range set generating polyno-mial is the following polynopolyno-mial:

(1.8) P (z) = Q(z) + c, where Q(z) = m X i=0 n X j=0 m i n j ₍₋₁₎i+j n + m + 1 − i − jz n+m+1−i−j_aj_bi_,

a 6= b, b 6= 0, c 6∈ {0, −Q(a), −Q(b), −Q(a)+Q(b)₂ }. Also, m, n are two integers such that m + n > 9, max{m, n} ≥ 3 and min{m, n} ≥ 2.

Remark 1.2. If we take a = 0 and b = 1 in (1.8), then we get the unique range set generating polynomial of degree at least 11. For details, see [5, 6].

2. Unique range sets with weight two

Let l be a non-negative integer or infinity. For a ∈ C ∪ {∞}, we denote by El(a; f ), the set of all a-points of f , where an a-point of multiplicity m is counted m times if m ≤ l and l + 1 times if m > l.

If for two non-constant meromorphic functions f and g, we have El(a; f ) = El(a; g),

then we say that f and g share the value a with weight l. Thus the IM and CM sharing respectively correspond to the weight 0 and ∞. This idea of weighted sharing was first introduced in ([18]).

Let S ⊂ C ∪ {∞}. We define Ef(S, l) as Ef(S, l) =

[

a∈S

El(a; f ),

where l is a non-negative integer or infinity. Clearly Ef(S) = Ef(S, ∞). Let l be a non-negative integer or infinity. A set S ⊂ C is called a unique range set for meromorphic (resp. entire) functions with weight l, in short, URSMl(resp. URSEl) if for any two non-constant meromorphic (resp. entire) functions f and g, the condition

Ef(S, l) = Eg(S, l) implies f ≡ g.

In the last few years, the notion of weighted sharing took place a major role in study of the uniqueness theory of meromorphic functions. As a result most of the existing unique range sets were relaxed to the unique range sets with weight two, but the cardinality of the respective unique range sets remained same when the sharing environment is relaxed from the CM sharing to the weighted sharing with weight two.

In this direction, in 2012, A. Banerjee and I. Lahiri made the following observations:

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Theorem 2.1 ([10]). Let P (z) = anzn+P m

j=2ajzj+ a0 be a polynomial of degree n, where n − m ≥ 3 and apam 6= 0 for some positive integer p with 2 ≤ p ≤ m and gcd(p, 3) = 1. Suppose further that S = {α1, α2, . . . , αn} is the set of all distinct zeros of P (z). Let k be the number of distinct zeros of the derivative P0(z). If n ≥ 2k + 7 (resp. 2k + 3), then the following statements are equivalent:

(i) P is a “uniqueness polynomial” for meromorphic (resp. entire) func-tion.

(ii) S is a URSM2 (resp. URSE2). (iii) S is a URSM (resp. URSE).

(iv) P is a “uniqueness polynomial in broad sense” for meromorphic (resp. entire) function.

To prove Theorem 2.1, the authors used the following lemma: Lemma 2.1 ([10], Lemma 2.1). Let P (z) = anzn +P

m

j=2ajzj + a0 be a polynomial of degree n, where n − m ≥ 3 and apam 6= 0 for some positive integer p with 2 ≤ p ≤ m and gcd(p, 3) = 1. Suppose that

1 P (f ) =

c0 P (g)+ c1,

where f and g are non-constant meromorphic functions and c0(6= 0), c1 are constants. If n ≥ 6, then c1= 0.

It is noted that in Lemma 2.1, the condition n−m ≥ 3 is necessary. Thus the polynomial considered in Theorem 2.1 is a specific polynomial (i.e., n − m ≥ 3, i.e., a gap after the n-th degree term).

The main observation of this paper is that the condition “n − m ≥ 3” is not necessary in order to show the equivalency between a unique range set with counting multiplicity and a unique range set with weight two.

Before going to state our main results, we need to introduce the deficiency functions ([17, 20]).

Let a ∈ C ∪ {∞}, we set

δ(a; f ) = 1 − lim sup r→∞

N (r, a; f ) T (r, f ) , Θ(a; f ) = 1 − lim sup

r→∞

N (r, a; f ) T (r, f ) .

The quantity δ(a; f ) is called the deficiency of the value a. Clearly 0 ≤ δ(a; f ) ≤ Θ(a; f ) ≤ 1.

Now, we state the main result of this paper.

Theorem 2.2. Let P (z) be a polynomial of degree n such that P (z) = a0(z − α1)(z − α2) · · · (z − αn); where αi 6= αj, 1 ≤ i, j ≤ n. Further suppose that S = {α1, α2, . . . , αn} is the set of all distinct zeros of P (z). Let k be the number

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of distinct zeros of the derivative P0(z). Let f and g be two non-constant meromorphic functions such that

Θ(∞; f ) + Θ(∞; g) +1

2min{δ(0, f ), δ(0, g)} >

2k + 6 − n

2 .

Then the following two statements are equivalent: (a) If Ef(S, 2) = Eg(S, 2), then f ≡ g. (b) If Ef(S) = Eg(S), then f ≡ g.

The following corollary is an immediate consequence of Theorem 2.2. Corollary 2.1. Let P (z) be a polynomial of degree n such that P (z) = a0(z − α1)(z − α2) · · · (z − αn); where αi 6= αj, 1 ≤ i, j ≤ n. Further suppose that S = {α1, α2, . . . , αn} is the set of all distinct zeros of P (z). Let k be the number of distinct zeros of the derivative P0(z). If n ≥ 2k + 7 (resp. 2k + 3), then the following two statements are equivalent:

(a) S is a URSM2 (resp. URSE2). (b) S is a URSM (resp. URSE).

Remark 2.1. Now, we consider the following polynomial:

(2.1) P (z) = zn+ azn−m+ b,

where a and b are two non-zero constants such that zn+ azn−m+ b = 0 has no multiple roots; m ≥ 2 (resp. 1), n ≥ 2m + 9 (resp. 2m + 5) are integers with n and n − m having no common factors.

Here k = m + 1 and this polynomial satisfies the assumptions of Corollary 2.1. Since the zero set of the polynomial gives URSM (resp. URSE) with 13 (resp. 7) elements ([24]), thus the zero set of the polynomial gives URSM2 (resp. URSE2) with 13 (resp. 7) elements.

Remark 2.2. The next polynomial was due to G. Frank and M. Reinders ([11]). (2.2) P (z) =(n − 1)(n − 2) 2 z n_{− n(n − 2)z}n−1₊n(n − 1) 2 z n−2_{− c,} where n ≥ 11 and c 6= 0, 1.

Here k = 2 and this polynomial satisfies the assumptions of Corollary 2.1. Since the zero set of the corresponding polynomial gives URSM (resp. URSE) with 11 (resp. 7) elements ([11]), thus the zero set of the respective polynomial gives URSM2 (resp. URSE2) with 11 (resp. 7) elements.

Remark 2.3. The polynomial described in the equation (1.8) satisfies the as-sumptions of Corollary 2.1. Here also, k = 2. Since the zero set of the polyno-mial gives URSM (resp. URSE) with 11 (resp. 7) elements ([8]), thus the zero set of the polynomial gives URSM2(resp. URSE2) with 11 (resp. 7) elements. Now, we explain some definitions and notations which are used to proceed further.

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(i) We denote by N (r, a; f |= 1) the counting function of simple a-points of f .

(ii) We denote by N (r, a; f |≤ m) (resp. N (r, a; f |≥ m) by the count-ing function of those a-points of f whose multiplicities are not greater (resp. less) than m where each a-point is counted according to its mul-tiplicity.

Similarly, N (r, a; f |≤ m) and N (r, a; f |≥ m) are the reduced counting func-tion of N (r, a; f |≤ m) and N (r, a; f |≥ m) respectively.

Definition 2.2 ([20]). Let f and g be two non-constant meromorphic functions such that f and g share a IM. Let z0 be an a-point of f with multiplicity p, an a-point of g with multiplicity q.

(i) We denote by NL(r, a; f ) the reduced counting function of those a-points of f and g where p > q.

(ii) We denote by N_E1)(r, a; f ) the counting function of those a-points of f and g where p = q = 1.

(iii) We denote by N(2_E(r, a; f ) the reduced counting function of those a-points of f and g where p = q ≥ 2.

In the same way, we can define NL(r, a; g), N 1)

E(r, a; g), N (2

E(r, a; g). When f and g share a with weight m, m ≥ 1 then

N_E1)(r, a; f ) = N (r, a; f |= 1).

Definition 2.3 ([20]). Let f , g share a value a IM. We denote by N∗(r, a; f, g) the reduced counting function of those a-points of f whose multiplicities differ from the multiplicities of the corresponding a-points of g. Clearly

N∗(r, a; f, g) ≡ N∗(r, a; g, f ) and N∗(r, a; f, g) = NL(r, a; f ) + NL(r, a; g). Proof of Theorem 2.2. The case (a)⇒(b) is obvious. So, we only prove the case (b)⇒(a).

Let f and g be two non-constant meromorphic functions share the set S = {α1, α2, . . . , αn}

with weight 2 and

Θ(∞; f ) + Θ(∞; g) +1

2min{δ(0, f ), δ(0, g)} >

2k + 6 − n

2 .

In this case, our claim is to show that f ≡ g. For that, we put P (z) = a0(z − α1)(z − α2) · · · (z − αn), and F (z) := 1 P (f (z)) and G(z) := 1 P (g(z)).

Let S(r) be any function S(r) : (0, ∞) → R satisfying S(r) = o(T (r, F ) + T (r, G)) for r → ∞ outside a set of finite Lebesgue measure.

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Let H(z) :=F 00_(z) F0_(z) − G00(z) G0_(z), and this function H was introduced by H. Fujimoto ([12]).

Now we consider two cases:

Case-I. First we assume that H 6≡ 0. It is given that

2k + 6 − 2Θ(∞; f ) − 2Θ(∞; g) − min{δ(0, f ), δ(0, g)} + < n, where is a small positive number.

Since H(z) can be expressed as H(z) =G 0_(z) F0_(z) F0_(z) G0_(z) 0 , so all poles of H are simple. Also, poles of H may occur at

(1) poles of F and G. (2) zeros of F0 and G0,

Now, by simple calculations, one can show that “simple poles” of F are the zeros of H. Thus

(2.3) N (r, ∞; F | = 1) = N (r, ∞; G| = 1) ≤ N (r, 0; H).

Now, using the lemma of logarithmic derivative and the first fundamental the-orem, (2.3) can be written as

(2.4) N (r, ∞; F | = 1) = N (r, ∞; G| = 1) ≤ N (r, ∞; H) + S(r). Let β1, β2, . . . , βk be the k-distinct zeros of P0(z). Since

F0(z) = −f

0_(z)P0_{(f (z))} (P (f (z)))2 , G

0_{(z) = −}g0(z)P0(g(z)) (P (g(z)))2

and f , g share S with weight 2, so by simple calculations, we can write N (r, ∞; H) (2.5) ≤ k X j=1 N (r, βj; f ) + N (r, βj; g) + N0(r, 0; f0) + N0(r, 0; g0) + N (r, ∞; f ) + N (r, ∞; g) + N∗(r, ∞; F, G),

where N0(r, 0; f0) denotes the reduced counting function of zeros of f0, which are not zeros ofQn

i=1(f − αi)Q k j=1(f − βj), similarly, N0(r, 0; g0) is defined. Since N (r, ∞; F | ≥ 2) + N0(r, 0, g0) + N∗(r, ∞; F, G) (2.6) ≤ N (r, 0; P (g)| ≥ 2) + N0(r, 0, g0) + N (r, 0; P (g)| ≥ 3) ≤ N (r, 0; g0),

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so, using Lemma 3 of ([21]), we get

N (r, ∞; F | ≥ 2) + N0(r, 0, g0) + N∗(r, ∞; F, G) (2.7)

≤ N (r, 0; g) + N (r, ∞; g) + S(r, g).

Put T (r) = max{T (r, f ), T (r, g)} and δ(0) = min{δ(0, f ), δ(0, g)}. Now, for any ε(> 0), using the second fundamental theorem and (2.4), (2.5) and (2.7), we have (n + k − 1)T (r, f ) (2.8) ≤ N (r, ∞; f ) + N (r, 0; P (f )) + k X j=1 N (r, βj; f ) − N0(r, 0; f0) + S(r, f ) ≤ 2N (r, ∞; f ) + N (r, ∞; g) + k X j=1 2N (r, βj; f ) + N (r, βj; g) + N (r, ∞; F | ≥ 2) + N0(r, 0; g0) + N∗(r, ∞; F, G) + S(r) ≤ 2N (r, ∞; f ) + 2N (r, ∞; g) + 2kT (r, f ) + kT (r, g) + N (r, 0; g) + S(r) ≤ (3k + 5 − 2Θ(∞; f ) − 2Θ(∞; g) − δ(0) + ε) T (r) + S(r). Similarly, (n + k − 1)T (r, g) (2.9) ≤ (3k + 5 − 2Θ(∞; g) − 2Θ(∞; f ) − δ(0) + ε) T (r) + S(r). Thus comparing (2.8) and (2.9), we have

(n + k − 1)T (r) (2.10)

≤ (3k + 5 − 2Θ(∞; g) − 2Θ(∞; f ) − δ(0) + ε) T (r) + S(r), which contradicts the assumption that

Θ(∞; f ) + Θ(∞; g) +1

2min{δ(0, f ), δ(0, g)} >

2k + 6 − n

2 .

Hence H ≡ 0.

Case-II. Next we assume that H ≡ 0. Then by integration, we have 1 P (f (z)) ≡ c0 P (g(z))+ c1, i.e., 1 a0(f − α1)(f − α2) · · · (f − αn) ≡ c0 a0(g − α1)(g − α2) · · · (g − αn) + c1, where c0is a non-zero complex constant. If z0is an αipoint of f of multiplicity m, then it is a pole of _{P (f (z))}1 of order m, hence it is a pole of _{P (g(z))}1 of order m, i.e., z0 is an αj point of g of order m for some j ∈ {1, 2, . . . , n}.

Thus f and g share the set S = {α1, α2, . . . , αn} in counting multiplicity. Since S is an URSM, so f ≡ g. This completes the proof.

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3. Unique range sets with weak weight three

Let l be a positive integer or infinity. For a ∈ C ∪ {∞}, we denote by El)(a; f ), the set of all a-points of f , whose multiplicities are not greater than l and each such a-points are counted according to its multiplicity.

If for two non-constant meromorphic functions f and g, we have El)(a; f ) = El)(a; g),

then we say that f and g share the value a with “weak-weight l”. Let S ⊂ C ∪ {∞}. We put

El)(S, f ) = [

a∈S

El)(a; f ), where l is a positive integer or infinity.

A set S ⊂ C is called a unique range set for meromorphic (resp. entire) func-tions with weak weight l, in short URSMl)(resp. URSEl)) if for any two non-constant meromorphic (resp. entire) functions f and g, El)(S, f ) = El)(S, g) implies f ≡ g.

In 2009, X. Bai, Q. Han and A. Chen ([4]) proved the following “weak-weighted” sharing version of Fujimoto’s Theorem:

Theorem 3.1 ([4]). In addition to the hypothesis of Theorem 1.1, further we suppose that l ≥ 3 is a positive integer or ∞.

If S is the set of zeros of P (z) and n > 2k + 6 (resp. n > 2k + 2), then S is a U RSMl) (resp. U RSEl)).

Using the concept of weighted sharing and weak-weighted sharing, Banerjee and Lahiri ([10]) gave some equivalence between the different notions of unique range sets and uniqueness polynomials as follows:

Theorem 3.2 ([10]). Let P (z) = anzn+P m

j=2ajzj+ a0 be a polynomial of degree n, where n − m ≥ 3 and apam 6= 0 for some positive integer p with 2 ≤ p ≤ m and gcd(p, 3) = 1. Suppose further that S = {α1, α2, . . . , αn} is the set of all distinct zeros of P (z). Let k be the number of distinct zeros of the derivative P0(z). If n ≥ 2k + 7 (resp. 2k + 3), then the following statements are equivalent:

(i) P is a “uniqueness polynomial” for meromorphic (resp. entire) func-tion.

(ii) S is a URSM3)(resp. URSE3)). (iii) S is a URSM (resp. URSE).

(iv) P is a “uniqueness polynomial in broad sense” for meromorphic (resp. entire) function.

Here, we also observed that to show the equivalence between the statements (ii) and (iii) in Theorem 3.2, one does not need the condition “n − m ≥ 3”. Now, we state our next result:

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Theorem 3.3. Let P (z) be a polynomial of degree n such that P (z) = a0(z − α1)(z − α2) · · · (z − αn); where αi 6= αj, 1 ≤ i, j ≤ n. Further suppose that S = {α1, α2, . . . , αn} is the set of all distinct zeros of P (z). Let k be the number of distinct zeros of the derivative P0(z). Let f and g be two non-constant meromorphic functions such that

Θ(∞; f ) + Θ(∞; g) +1

2min{δ(0, f ), δ(0, g)} >

2k + 6 − n

2 .

Then the following two statements are equivalent: (a) If E3)(S, f ) = E3)(S, g), then f ≡ g. (b) If Ef(S) = Eg(S), then f ≡ g.

The proof of this theorem is similar to the proof of Theorem 2.2. So we omit the details.

Corollary 3.1. Let P (z) be a polynomial of degree n such that P (z) = a0(z − α1)(z − α2) · · · (z − αn); where αi 6= αj, 1 ≤ i, j ≤ n. Further suppose that S = {α1, α2, . . . , αn} is the set of all distinct zeros of P (z). Let k be the number of distinct zeros of the derivative P0(z). If n ≥ 2k + 7 (resp. 2k + 3), then the following two statements are equivalent:

(a) S is a URSM3)(resp. URSE3)). (b) S is a URSM (resp. URSE).

The next result is obvious in view of Theorem 2.2 and Theorem 3.3: Corollary 3.2. Let P (z) be a polynomial of degree n such that P (z) = a0(z − α1)(z − α2) · · · (z − αn); where αi 6= αj, 1 ≤ i, j ≤ n. Further suppose that S = {α1, α2, . . . , αn} is the set of all distinct zeros of P (z). Let k be the number of distinct zeros of the derivative P0(z). Let f and g be two non-constant meromorphic functions such that

Θ(∞; f ) + Θ(∞; g) +1

2min{δ(0, f ), δ(0, g)} >

2k + 6 − n

2 .

Then the following statements are equivalent: (a) If Ef(S) = Eg(S), then f ≡ g. (b) If Ef(S, 2) = Eg(S, 2), then f ≡ g.

(c) If E3)(S, f ) = E3)(S, g), then f ≡ g.

4. Functions sharing two sets

In connection to the Gross’s question (Question 1.1), in 1994, H. X. Yi ([23]) gave the existence of two finite sets S1 (with 5 elements) and S2 (with one element) such that if any two non-constant entire functions f and g satisfying the condition E(Sj, f ) = E(Sj, g) for j = 1, 2, then f ≡ g.

Later in 1998, the same author ([25]) proved that there exist two finite sets S1 (with 3 elements) and S2 (with one element) such that any two non-constant entire functions f and g satisfying the condition E(Sj, f ) = E(Sj, g) for j = 1, 2, then f ≡ g.

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In the same paper ([25]), another nice observation was made.

Theorem 4.1 ([25]). If S1and S2are two sets of finite distinct complex num-bers such that any two entire functions f and g satisfying Ef(Sj) = Eg(Sj) for j = 1, 2, must be identical, then max{](S1), ](S2)} ≥ 3, where ](S) denotes the cardinality of the set S.

Thus for the uniqueness of two entire functions when they share two sets, it is clear that the smallest cardinalities of S1 and S2 are 1 and 3 respectively.

Later, the Gross’ Question for meromorphic functions was also introduced in the literature as follows:

Question 4.1 ([23]). Can one find two finite sets Sj(j = 1, 2) such that if two non-constant meromorphic functions f and g share them, then f ≡ g?

In 1994, H. X. Yi ([23]) completely answered Question 4.1 by giving the existence of two finite sets S1 (with 9 elements) and S2 (with 2 elements) such that if any two non-constant meromorphic functions f and g satisfying the condition E(Sj, f ) = E(Sj, g) for j = 1, 2, then f ≡ g.

In this direction, in 2012, B. Yi and Y. H. Li ([26]) provided a significant result. They proved that there exist two finite sets S1 (with 5 elements) and S2(with 2 elements) such that if any two non-constant meromorphic functions f and g satisfying the condition E(Sj, f ) = E(Sj, g) for j = 1, 2, then f ≡ g.

The motivation of writing this section is to answer Question 4.1 by giving the existence of two generic sets S1 and S2 for meromorphic functions such that if any two non-constant meromorphic functions f and g satisfying the condition E(Sj, f ) = E(Sj, g) for j = 1, 2, then f ≡ g.

Suppose

(4.1) P (z) = a0(z − α1)(z − α2) · · · (z − αn), where αi6= αj, 1 ≤ i, j ≤ n; a06= 0. Further suppose that (4.2) P0(z) = b0(z − β1)q1(z − β2)q2· · · (z − βk)qk

satisfying the assumption (this property was introduced by H. Fujimoto ([12])) that

(4.3) P (βls) 6= P (βlt) (1 ≤ ls< lt≤ k).

Now, we state our main two theorems of this section:

Theorem 4.2. Let P (z) be a “uniqueness polynomial” of the form (4.1) sat-isfying the condition (4.3). Further suppose that S1 = {α1, α2, . . . , αn} and S2= {β1, β2, . . . , βk}.

If two non-constant meromorphic (resp. entire) functions f and g share the set S1 with weight two and S2 IM, k ≥ 3 and n ≥ k + 7 (resp. k + 3), then f ≡ g.

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Theorem 4.3. Let P (z) be a “uniqueness polynomial” of the form (4.1) sat-isfying the condition (4.3). Further suppose that S1 = {α1, α2, . . . , αn} and S2= {β1, β2, . . . , βk}.

Moreover, assume that k ≥ 2 and P0(z) have no simple zeros. If two non-constant meromorphic (resp. entire) functions f and g share the set S1 with weight 3 and S2 IM, and n ≥ max{10 − 2k, 5} (resp. 5), then f ≡ g.

Example 4.1. Let P (z) = (n − 1)(n − 2) 2 z n_{− n(n − 2)z}n−1₊n(n − 1) 2 z n−2_{− c,} where c ∈ C \ {0,1 2, 1} and n ≥ 6.

Under these suppositions, P (z) must be a “uniqueness polynomial” satisfy-ing the condition (4.3) (Theorem 1.2, [9]). Suppose that

S1= z : (n − 1)(n − 2) 2 z n_{− n(n − 2)z}n−1₊n(n − 1) 2 z n−2_{− c = 0}

and S2 = {0, 1}. Then by Theorem 4.3, if two non-constant meromorphic functions f and g share S1with weight 3 and S2IM, then f ≡ g.

To prove the above two theorems, we need the following lemma:

Lemma 4.1 ([12, Proposition 7.1]). Let P (z) be a polynomial of degree n ≥ 5 and of the form (4.1) satisfying the condition (4.3). Suppose that

1 P (f ) =

c0 P (g)+ c1,

where f and g are non-constant meromorphic functions and c0(6= 0), c1 are constants. If k ≥ 3, or if k = 2 and P0(z) have no simple zeros, then c1= 0. Proof of Theorem 4.2. Since f and g share S2 IM, so

k X j=1 N (r, βj; f ) = k X j=1 N (r, βj; g). Now, we put F (z) := 1 P (f (z)) and G(z) := 1 P (g(z)).

Let H(z) :=F 00_(z) F0_(z) − G00(z) G0_(z). Now, we consider two cases:

Case-I. First we assume that H 6≡ 0. Since H(z) can be expressed as H(z) =G 0_(z) F0_(z) F0_(z) G0_(z) 0 ,

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so all poles of H are simple. Also, poles of H may occur at (1) poles of F and G.

(2) zeros of F0 and G0.

But using the Laurent series expansion of H, it is clear that “simple poles” of F (hence, that of G) is a zero of H. Thus

(4.4) N (r, ∞; F | = 1) = N (r, ∞; G| = 1) ≤ N (r, 0; H).

Using the lemma of logarithmic derivative and the first fundamental theorem, (4.4) can be written as (4.5) N (r, ∞; F | = 1) = N (r, ∞; G| = 1) ≤ N (r, ∞; H) + S(r). Since F0(z) = −f 0_(z)P0_{(f (z))} (P (f (z)))2 and G 0_{(z) = −}g0(z)P0(g(z)) (P (g(z)))2 , and f , g share (S1, 2) and (S2, 0), so by simple calculations, we can write

N (r, ∞; H) ≤ k X j=1 N (r, βj; f ) + N0(r, 0; f0) + N0(r, 0; g0) (4.6) + N (r, ∞; f ) + N (r, ∞; g) + N∗(r, ∞; F, G),

where N0(r, 0; f0) denotes the reduced counting function of zeros of f0, which are not zeros of Qn

i=1(f − αi) Qk

j=1(f − βj); similarly, N0(r, 0; g

0_{) is defined.} Now, using the second fundamental theorem and (4.5), (4.6), we have

(n + k − 1) (T (r, f ) + T (r, g)) (4.7) ≤ N (r, ∞; f ) + N (r, ∞; g) + N (r, 0; P (f )) + N (r, 0; P (g)) + k X j=1 N (r, βj; f ) + N (r, βj; g) − N0(r, 0; f0) − N0(r, 0; g0) + S(r) ≤ 2 N (r, ∞; f ) + N (r, ∞; g) + k X j=1 2N (r, βj; f ) + N (r, βj; g) + N (r, ∞; F | ≥ 2) + N (r, ∞; G) + N∗(r, ∞; F, G) + S(r). Noting that N (r, ∞; F ) −1 2N (r, ∞; F | = 1) + 1 2N∗(r, ∞; F, G) ≤ 1 2N (r, ∞; F ), N (r, ∞; G) −1 2N (r, ∞; G| = 1) + 1 2N∗(r, ∞; F, G) ≤ 1 2N (r, ∞; G). Thus (4.7) can be written as

(n + k − 1) (T (r, f ) + T (r, g)) (4.8) ≤ 2 N (r, ∞; f ) + N (r, ∞; g) + 3 2k + n 2 (T (r, f ) + T (r, g)) + S(r),

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which contradicts the assumption n ≥ k + 7 (resp. k + 3). Thus H ≡ 0. Case-II. Next we assume that H ≡ 0. Then by integration, we have

1 P (f (z)) ≡

c0

P (g(z))+ c1, where c0(6= 0), c1 are constants.

Since n ≥ 5 and k ≥ 3, thus by applying Lemma 4.1 and noting the assump-tion that P (z) is a “uniqueness polynomial”, we have

f ≡ g.

This completes the proof.

Proof of Theorem 4.3. Since f and g share S2 IM, so k X j=1 N (r, βj; f ) = k X j=1 N (r, βj; g). Now, we put F (z) := 1 P (f (z)) and G(z) := 1 P (g(z)).

Let H(z) :=F 00_(z) F0_(z) − G00(z) G0_(z). Now, we consider two cases:

Case-I. First we assume that H 6≡ 0. Now, proceeding as Case-I of Theorem 4.2, we have from (4.7) that

(n + k − 1) (T (r, f ) + T (r, g)) (4.9) ≤ 2 N (r, ∞; f ) + N (r, ∞; g) + k X j=1 2N (r, βj; f ) + N (r, βj; g) + N (r, ∞; F | ≥ 2) + N (r, ∞; G) + N∗(r, ∞; F, G) + S(r). Since f and g share the set S1 with weight 3, so

N (r, ∞; F ) + N (r, ∞; G) − N (r, ∞; F | = 1) +5

2N∗(r, ∞; F, G) ≤ 1

2(N (r, ∞; F ) + N (r, ∞; G)). Thus (4.9) can be written as

(n + k − 1) (T (r, f ) + T (r, g)) (4.10) ≤ 2 N (r, ∞; f ) + N (r, ∞; g) + 3 k X j=1 N (r, βj; f )

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+1

2(N (r, ∞; F ) + N (r, ∞; G)) − 3

2N∗(r, ∞; F, G) + S(r). Now, let us consider the following function

ϕ(z) := F 0_(z) F (z) −

G0(z) G(z). Next we consider two cases:

Subcase-I. Assume that ϕ 6≡ 0. Thus all poles of ϕ are simple. Also, poles of ϕ may occur at

(1) poles of F and G, (2) zeros of F and G.

Since P0(z) have no simple zeros, thus using the first fundamental theorem and the lemma of logarithmic derivative, we have

2 k X j=1 N (r, βj; f ) ≤N (r, 0; ϕ) ≤T (r, ϕ) + O(1) ≤N (r, ∞; ϕ) + S(r, F ) + S(r, G) ≤N (r, ∞; f ) + N (r, ∞; g) + N∗(r, ∞; F, G) + S(r). Thus (4.10) can be written as

(n

2 + k − 1) (T (r, f ) + T (r, g)) ≤ 7

2 N (r, ∞; f ) + N (r, ∞; g) + S(r), (4.11)

which is impossible if f and g both are entire functions. Also, this is impossible if f and g both are meromorphic functions and n ≥ 10 − 2k. Thus ϕ ≡ 0. Subcase-II. Next we assume that ϕ ≡ 0. Then on integration, we have

F ≡ AG,

where A is a non-zero constant, which is impossible as H 6≡ 0.

Case-II. Now we consider the case H ≡ 0. Then by integration, we have 1

P (f (z)) ≡ c0

P (g(z))+ c1, where c0(6= 0), c1 are constants.

Since n ≥ 5, k ≥ 2 and P0(z) have no simple zeros, thus, applying Lemma 4.1 and noting the assumption that P (z) is a “uniqueness polynomial”, we have

f ≡ g.

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5. Some observations

The natural query would be whether there exist different classes of unique range sets, but in this direction the number of results are not sufficient. Re-cently, V. H. An and P. N. Hoa ([3]) exhibited a new class of unique range set for meromorphic functions. The unique range set is the zero set of the following polynomial:

(5.1) P (z) = zn+ (az + b)n+ c,

where n ≥ 25 is an integer, a, b, c ∈ C \ {0} with c 6= bd ad, a

2d _{6= 1, c 6= a}d_bd_, c 6= (−1)_a2ddbd, c 6= (−1)

d_bd_{. Also, it was assumed that P (z) has only simple} zeros.

This URSM has 25 elements but in literature there exist URSM with 11 elements. Also, it was proved that any URSM (resp. URSE) must contain at least six (resp. five) [see, Theorem 10.59 (resp. Theorem 10.72), ([20])] elements. So, the challenging work is to exhibit URSM (resp. URSE) with elements ≤ 11 (resp. 7).

Now, we discuss a method of P. Li and C. C. Yang (page 448, ([19])): Let S = {α1, α2, . . . , αn} be a set with finite distinct elements of C. Also, let α(6= 0) and β be two complex constants. If S is a unique range set, then the set T = {αα1+ β, αα2+ β, . . . , ααn+ β} is also a unique range set.

If f and g are two meromorphic functions sharing T CM, then (f − (αα1+ β))(f − (αα2+ β)) · · · (f − (ααn+ β)) = h(g − (αα1+ β))(g − (αα2+ β)) · · · (g − (ααn+ β)),

where h is a meromorphic function whose zeros come from the poles of g and the poles come from the poles of f . Thus

f − β α − α1 f − β α − α2 . . . f − β α − αn = h g − β α − α1 g − β α − α2 . . . g − β α − αn .

Thus f −β_α and g−β_α share S CM. So, f −β_α ≡g−β_α , i.e., f ≡ g. So, T is a URSM. Remark 5.1. Since the examples of unique range sets are few in numbers, thus this method (page 448, [19]) helps us to construct new class of unique range sets. For examples,

(i) The zero set of the following polynomial gives a new class of URSM (resp. URSE) with 13 (resp. 7) elements:

P (z) = (z − β)n+ a(z − β)n−m+ b,

where β ∈ C, a and b are two non-zero constants such that zn ₊ azn−m_{+ b = 0 has no multiple root. Also, m ≥ 2 (resp. 1), n > 2m + 8} (resp. 2m+4) are integers with n and n−m having no common factors.

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(ii) The zero set of the following polynomial gives a new class of URSM with 11 elements: P (z) = (n − 1)(n − 2) 2 (z − β) n_{− n(n − 2)(z − β)}n−1₊n(n − 1) 2 (z − β) n−2_{− c,} where n ≥ 11 and c 6= 0, 1, β ∈ C.

We have seen from the equation (1.1) that the zero set of the polynomial P (z) = zn+ zn−1+ 1

gives a URSE with n(≥ 7) elements. Now, znP 1

z

= zn+ z + 1. Again, the zero set of the polynomial zn_{P (}1

z) = z

n_{+ z + 1 gives a URSE with} n(≥ 7) elements (Theorem 10.57, [20]).

Next, we will see that the equation (1.3) gives that the zero set of the polynomial P (z) = (n − 1)(n − 2) 2 z n − n(n − 2)zn−1+n(n − 1) 2 z n−2 −c 2, where c 6= 0, 1, 2 is a URSM with n(≥ 11) elements. Now,

znP 1 z = −1 2 cz n − n(n − 1)z2+ 2n(n − 2)z − (n − 1)(n − 2) . From the equation (1.4), the zero set of the polynomial zn_{P (}1

z) also gives a URSM with n(≥ 11) elements.

Thus the following question is obvious:

Question 5.1. Let P (z) be a non-constant polynomial of degree n, having simple zeros. What are the characterizations of the polynomial P (z) such that if the zero set of the polynomial P (z) forms a unique range set, then the zero set of polynomial zn_{P (}1

z) must form a unique range set? References

[1] T. C. Alzahary, Meromorphic functions with weighted sharing of one set, Kyungpook Math. J. 47 (2007), no. 1, 57–68.

[2] T. T. H. An, Unique range sets for meromorphic functions constructed without an injectivity hypothesis, Taiwanese J. Math. 15 (2011), no. 2, 697–709. https://doi.org/ 10.11650/twjm/1500406229

[3] V. H. An and P. N. Hoa, A new class of unique range sets for meromorphic functions, Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 109–116.

[4] X. Bai, Q. Han, and A. Chen, On a result of H. Fujimoto, J. Math. Kyoto Univ. 49 (2009), no. 3, 631–643. https://doi.org/10.1215/kjm/1260975043

[5] A. Banerjee, A new class of strong uniqueness polynomials satisfying Fujimoto’s con-dition, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 465–474. https://doi.org/10. 5186/aasfm.2015.4025

[6] A. Banerjee and B. Chakraborty, A new type of unique range set with deficient values, Afr. Mat. 26 (2015), no. 7-8, 1561–1572. https://doi.org/10.1007/s13370-014-0305-4

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[7] , Further results on the uniqueness of meromorphic functions and their derivative counterpart sharing one or two sets, Jordan J. Math. Stat. 9 (2016), no. 2, 117–139. [8] , On some sufficient conditions of strong uniqueness polynomials, Adv. Pure

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[10] A. Banerjee and I. Lahiri, A uniqueness polynomial generating a unique range set and vice versa, Comput. Methods Funct. Theory 12 (2012), no. 2, 527–539. https://doi. org/10.1007/BF03321842

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[12] H. Fujimoto, On uniqueness of meromorphic functions sharing finite sets, Amer. J. Math. 122 (2000), no. 6, 1175–1203.

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[14] F. Gross, On the distribution of values of meromorphic functions, Trans. Amer. Math. Soc. 131 (1968), 199–214. https://doi.org/10.2307/1994690

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[17] W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.

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[19] P. Li and C.-C. Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J. 18 (1995), no. 3, 437–450. https://doi.org/10.2996/kmj/ 1138043482

[20] C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, Mathematics and its Applications, 557, Kluwer Academic Publishers Group, Dordrecht, 2003. [21] H. X. Yi, Uniqueness of meromorphic functions and a question of C. C. Yang,

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[24] , Unicity theorems for meromorphic or entire functions. III, Bull. Austral. Math. Soc. 53 (1996), no. 1, 71–82. https://doi.org/10.1017/S0004972700016737

[25] , On a question of Gross concerning uniqueness of entire functions, Bull. Austral. Math. Soc. 57 (1998), no. 2, 343–349. https://doi.org/10.1017/S0004972700031701 [26] B. Yi and Y. H. Li, Uniqueness of meromorphic functions that share two sets with CM,

Acta Math. Sinica (Chin. Ser.) 55 (2012), no. 2, 363–368. Bikash Chakraborty

Department of Mathematics

Ramakrishna Mission Vivekananda Centenary College West Bengal 700 118, India

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Jayanta Kamila

Email address: kamilajayanta@gmail.com Amit Kumar Pal

Department of Mathematics University of Kalyani West Bengal 741 235, India

Email address: mail4amitpal@gmail.com Sudip Saha